# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Theory of fractional hybrid differential equations. (English) Zbl 1228.45017
Summary: We develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order $0. An existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.
##### MSC:
 45K05 Integro-partial differential equations 34A08 Fractional differential equations
##### References:
 [1] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equation, (1993) [2] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) [3] Podlubny, I.: Fractional differential equations, mathematics in science and engineering, (1999) [4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integral and derivative. Theory and applications, (1993) · Zbl 0818.26003 [5] Li, Q.; Sun, S.: On the existence of positive solutions for initial value problem to a class of fractional differential equation, , 886-889 (2010) [6] Li, Q.; Sun, S.; Zhang, M.; Zhao, Y.: On the existence and uniqueness of solutions for initial value problem of fractional differential equations, J. univ. Jinan 24, 312-315 (2010) [7] Q. Li, S. Sun, Z. Han, Y. Zhao, On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations, in: 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao, 2010, pp. 452–457. [8] Zhang, M.; Sun, S.; Zhao, Y.; Yang, D.: Existence of positive solutions for boundary value problems of fractional differential equations, J. univ. Jinan 24, 205-208 (2010) [9] Zhao, Y.; Sun, S.: On the existence of positive solutions for boundary value problems of nonlinear fractional differential equations, , 682-685 (2010) [10] Y. Zhao, S. Sun, Z. Han, M. Zhang, Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations, in: 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao, 2010, pp. 480–485. [11] Zhao, Y.; Sun, S.; Han, Z.; Li, Q.: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. nonlinear sci. Numer. simul. 16, No. 4, 2086-2097 (2011) · Zbl 1221.34068 · doi:10.1016/j.cnsns.2010.08.017 [12] Zhao, Y.; Sun, S.; Han, Z.; Li, Q.: Positive solutions to boundary value problems of nonlinear fractional differential equations, Abst. appl. Anal. 2011, 1-16 (2011) [13] Zhao, Y.; Sun, S.; Han, Z.; Zhang, M.: Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. math. Comput. 217, No. 16, 6950-6958 (2011) · Zbl 1227.34011 · doi:10.1016/j.amc.2011.01.103 [14] Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear anal. TMA 71, No. 7–8, 2724-2733 (2009) · Zbl 1175.34082 · doi:10.1016/j.na.2009.01.105 [15] Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal. TMA 71, No. 7–8, 3249-3256 (2009) · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202 [16] Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. RWA 11, 4465-4475 (2010) [17] Wang, J.; Zhou, Y.: A class of fractional evolution equations and optimal controls, Nonlinear anal. RWA 12, 262-272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013 [18] Agarwal, R. P.; Zhou, Y.; He, Y.: Existence of fractional neutral functional differential equations, Comput. math. Appl. 59, No. 3, 1095-1100 (2010) · Zbl 1189.34152 · doi:10.1016/j.camwa.2009.05.010 [19] Li, C. F.; Luo, X. N.; Zhou, Y.: Existence of positive solutions of boundary value problem for fractional differential equations, Comput. math. Appl. 59, No. 3, 1363-1375 (2010) · Zbl 1189.34014 · doi:10.1016/j.camwa.2009.06.029 [20] Diethelm, K.: The analysis of fractional differential equations, (2010) [21] Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. TMA 69, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042 [22] Lakshmikantham, V.; Vatsala, A. S.: Theory of fractional differential inequalities and applications, Commun. appl. Anal. 11, 395-402 (2007) · Zbl 1159.34006 [23] Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal. TMA 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025 [24] Lakshmikantham, V.; Devi, J. V.: Theory of fractional differential equations in Banach space, Eur. J. Pure appl. Math. 1, 38-45 (2008) · Zbl 1146.34042 · doi:http://www.ejpam.com/ejpam/index.php/ejpam/article/view/84 [25] Kilbas, A. A.; Srivastava, H. H.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) [26] Caputo, M.: Linear models of dissipation whose Q is almost independent, II, Geophy. J. Roy. astronom. 13, 529-539 (1967) [27] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194 [28] Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solution, Appl. math. Comput. 154, 621-640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2 [29] Dhage, B. C.: On $\alpha$-condensing mappings in Banach algebras, Math. student 63, 146-152 (1994) · Zbl 0882.47033 [30] Dhage, B. C.; Lakshmikantham, V.: Basic results on hybrid differential equations, Nonlinear anal. Hybrid 4, 414-424 (2010) · Zbl 1206.34020 · doi:10.1016/j.nahs.2009.10.005 [31] Dhage, B. C.: A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear funct. Anal. appl. 8, 563-575 (2004) · Zbl 1067.47070 [32] Dhage, B. C.: Fixed point theorems in ordered Banach algebras and applications, Panamer. math. J. 9, No. 4, 93-102 (1999) · Zbl 0964.47026 [33] Lakshmikantham, V.; Leela, S.: Differential and integral inequalities, (1969) · Zbl 0177.12403 [34] Dhage, B. C.: On a fixed point theorem in Banach algebras with applications, Appl. math. Lett. 18, 273-280 (2005) · Zbl 1092.47045 · doi:10.1016/j.aml.2003.10.014 [35] Heikkilä, S.; Lakshmikantham, V.: Monotone iterative technique for nonlinear discontinues differential equations, (1994)